LIke I said, I don't want to start an argument here, but I would say that the wide aperture on the S100 will outshine an old kit lens on a Rebel. Hence you don't need to use the high ISO.

The S100 lens is 1.5 stops faster at the wide end, and is 1/3 stop slower at the telephoto end vs. typical kit lenses. The T3i captures more detail and roughly the same amount of noise at ISO 3200 that the S100 does at ISO 400, a 3 stop difference. Because there's more detail you have greater room for NR meaning the practical difference is probably more like 4 stops. Now add in AF performance and shutter delay...

I don't know what kind of "gym action" this person hopes to capture or what the gym lighting conditions will be like. If the action requires solid AF tracking, an entry level DSLR is the minimum. If not, then the EOS M could be a good choice.

Personally I would recommend a used/refurbished entry level DSLR body and a fast prime at the required focal length if at all possible on that budget.

The Canon S100 is a great little camera and surprisingly clean at 1600 for a P&S. But, if action is the priority, then DSLT is the only answer. I'd look at a refurbished T4i, T3i, or T2i through Canon's refubish store.